SIMPLIFIED FINITE DIFFERENCE THERMAL LATTICE ...azwadi/pdf_publication/ISI3.pdfSeptember 3, 2008...
Transcript of SIMPLIFIED FINITE DIFFERENCE THERMAL LATTICE ...azwadi/pdf_publication/ISI3.pdfSeptember 3, 2008...
-
September 3, 2008 10:0 WSPC/140-IJMPB 04861
International Journal of Modern Physics BVol. 22, No. 22 (2008) 3865–3876c© World Scientific Publishing Company
SIMPLIFIED FINITE DIFFERENCE THERMAL LATTICE
BOLTZMANN METHOD
C. S. NOR AZWADI
Faculty of Mechanical Engineering, Universiti Teknologi Malaysia,
81310 UTM Skudai, Johor, Malaysia
T. TANAHASHI
Department of Mechanical Engineering, Keio University,
Yokohama, Kanagawa, Japan
Received 5 November 2007
In this paper, a well-known finite difference technique is combined with thermal latticeBoltzmann method to solve 2-dimensional incompressible thermal fluid flow problems. Asmall number of microvelocity components are applied for the calculation of temperaturefield. The combination of finite difference with lattice Boltzmann method is found to bean efficient and stable approach for the simulation at high Rayleigh number of naturalconvection in a square cavity.
Keywords: Thermal lattice Boltzmann; finite difference; FDLBM; natural convectionflow.
1. Introduction
The lattice Boltzmann method (LBM) is considered as an alternative approach to
the well-known finite difference, finite element, and finite volume techniques for
solving the Navier-Stokes equations. LBM evolved from Lattice Gas Automata,1
simulates fluid flows by tracking the evolution of the single-particle distribution.
Although as a newcomer in numerical scheme, the lattice Boltzmann approach has
found recent successes in a host of fluid dynamical problems, including flows in
porous media, magnetohydrodynamics, immiscible fluids, and turbulence. How-
ever, the simulation of flows with heat transfer turned out to be much more
difficult.
In general, the current thermal lattice Boltzmann models fall into three cate-
gories: the multi-speed approach,2 the passive scalar approach,3 and the thermal
energy distribution model proposed by He et al.4 The multi-speed approach uses
3865
-
September 3, 2008 10:0 WSPC/140-IJMPB 04861
3866 C. S. Nor Azwadi & T. Tanahashi
the same distribution function in defining the macroscopic velocity, pressure, and
temperature. In addition to mass and momentum, in order to preserve the kinetic
energy in the collision on each lattice point, this model requires more variations
of speed than those of the isothermal model and equilibrium distribution function
usually include higher order velocity terms. However, this model is reported to
suffer severe numerical instability, and is not computationally efficient.5
In the passive scalar model, the flow fields (velocity and density) and the tem-
perature are represented by two different distribution functions. The macroscopic
temperature is assumed to satisfy the same evolution equation as a passive scale,
which is advected by the flow velocity, but does not affect the flow field. It has
been shown that the passive scalar model has better numerical stability than the
multi-speed model.5
He et al.4 in their model introduce the internal energy density distribution func-
tion, which can be derived from the Boltzmann equation. This model is shown to
be a suitable model for simulating real thermal problems. However, the complicated
gradient operator term appears in the evolution equation and thus the simplicity
property of the lattice Boltzmann scheme has been lost.6
The conventional isothermal and thermal lattice Boltzmann models, however,
only give second-order accuracy in space and time. Since Luo and He7,8 and Abe9
demonstrated that the lattice Boltzmann equation is a discretized form of the con-
tinuous Boltzmann equation and the discretization of physical space is not coupled
with the discretization of momentum space, any standard numerical technique can
serve the purpose of solving the discrete Boltzmann equation. It is not surprising
that the well-known finite difference has being introduced in order to improve the
accuracy of isothermal and thermal LBM.
The first finite difference LBM (FDLBM) was due to Reider and Sterling,10
and was examined by Cao et al.11 in more detail. FDLBM was further extended
to curvilinear coordinates with non-uniform grids by Mei and Shyy.12 The study of
FDLBM is still in progress.13–15 However, there are still no evidence of combination
of finite difference with thermal lattice Boltzmann.
The purpose of this paper is to improve the earlier proposed double-distribution
function thermal lattice Boltzmann method (TLBM).16 Although this model has
successfully simulated the natural convection problem to a certain degree with
low computational cost, it is limited for the simulation at low Rayleigh numbers.
However, for real thermal engineering applications, the value of Rayleigh numbers
could be achieved up to 106. In this paper, we apply the finite difference technique to
solve the advection term in the governing equations of double-distribution function
TLBM. The combination of finite difference with TLBM (FDTLBM) contributes
in allowing us to increase the accuracy both in time and space where the high
order accuracy is crucial for the simulation of natural convection at high Rayleigh
numbers.
-
September 3, 2008 10:0 WSPC/140-IJMPB 04861
Finite Difference Thermal Lattice Boltzmann Method 3867
2. Double-Distribution Function TLBM
Following the double-distribution function approach proposed by He4 and Nor
Azwadi and Takahaski,16 the governing equations for these two functions are
∂f
∂t+ c
∂f
∂x= − 1
τv(f − f eq) + Ff , (1)
∂g
∂t+ c
∂g
∂x= − 1
τc(g − geq) , (2)
where the density distribution function f = f(x, c, t) is used to simulate the
density and velocity fields, and the internal energy density distribution function
g = g(x, c, t) is used to simulate the macroscopic temperature field. The macro-
scopic variables, such as the density ρ, velocity u, and temperature T can be eval-
uated as the moment to the distribution function
ρ =
∫
fdc , ρu =
∫
cfdc , ρT =
∫
gdc . (3)
f eq and geq in Eqs. (1) and (2) are the equilibrium distribution function for density
and internal energy, respectively, and is given by
f eq = ρ(1
2πRT)D/2 exp
{
− c2
2RT
} [
1 +c · uRT
+(c · u)22(RT )2
− u2
2RT
]
, (4)
geq = ρT
(
1
2πRT
)D/2
exp
{
− c2
2RT
}
[
1 +c · uRT
]
. (5)
Equation (5) is obtained by assuming that at low Mach number flow (incompressible
flow), the higher order of macroscopic velocity and viscous heat dissipation can be
neglected.6 It has also been proved17 that the above simplification does not alter
the corresponding macroscopic equation of energy. The only change is the value
of the constant parameter in the thermal conductivity, which can be absorbed by
manipulating the parameter τc.
We have also recently shown that the discretized equilibrium distribution func-
tion for both density and internal energy density distribution function can be ob-
tained by applying the Gauss–Hermite quadrature procedure for the calculation of
f eq and geq velocity moments. As a result, a 2-dimensional 9-velocity, D2Q9 lat-
tice model, as shown in Fig. 1(left), is obtained, and the corresponding discretized
equilibrium density distribution function is given by
f eqi = ρωi
[
1 + 3c · uc2
+9(c · u)2
2c4− 3u
2
2c2
]
, (6)
where c =√
3RT and the weights are ω1 = 4/9, ω2 = ω3 = ω4 = ω5 = 1/9,
and, ω6 = ω7 = ω8 = ω9 = 1/36. While the lattice type for the energy model
is 2-dimensional 4-velocity, D2Q4 lattice model shown in Fig. 1(right), and the
-
September 3, 2008 10:0 WSPC/140-IJMPB 04861
3868 C. S. Nor Azwadi & T. Tanahashi
5c
1c
8c
6c
7c
4c
3c
2c
0c
1c
4c
3c
2c
Fig. 1. Lattice structure for D2Q9 (left) and D2Q4 (right).
corresponding discretized internal energy density equilibrium distribution function
is given by16
geq1,2,3,4 =1
4ρT
[
1 +c · uc2
]
. (7)
Through a multiscaling expansion, the mass and momentum equation can be
derived from D2Q9 and temperature equation from D2Q4 as below
∇ · u = 0 , (8)
∂u
∂t+ u∇ · u = −1
ρ∇p + ν∇2u , (9)
∂T
∂t+ ∇ · (uT ) = χ∇2T . (10)
The viscosity and thermal diffusivity in these models are related to the time
relaxations as below
ν =1
3τv , (11)
χ = τc . (12)
3. Finite Difference Thermal Lattice Boltzmann Method
(FDTLBM)
The temporal discretization is obtained using second-order Runge–Kutta (modified)
Euler method. The time evolution of particle distributions is then derived by
fn+ 1
2
i = fni +
∆t
2
[
−ci · ∇fni −1
τv(fni − f eq,ni )
]
, (13)
fn+1i = fni + ∆t
[
−ci · ∇fn+1
2
i −1
τv
(
fn+ 1
2
i − feq,n+ 1
2
i
)
]
. (14)
-
September 3, 2008 10:0 WSPC/140-IJMPB 04861
Finite Difference Thermal Lattice Boltzmann Method 3869
The third-order upwind scheme (UTOPIA) was applied to calculate the advec-
tion term in Eq. (1) as below
cix∂xfi = cixfi(x + 2∆x, y) − 2fi(x + ∆x, y) + 9fi(x, y)
6∆x
+ cix−10fi(x − ∆x, y) + 2fi(x − 2∆x, y)
6∆x, cix > 0 , (15)
cix∂xfi = cix−fi(x − 2∆x, y) + 2fi(x − ∆x, y) − 9fi(x, y)
6∆x
+ cix10fi(x + ∆x, y) − 2fi(x + 2∆x, y)
6∆x, cix < 0 , (16)
ciy∂yfi = ciyfi(x, y + 2∆y) − 2fi(x, y + ∆y) + 9fi(x, y)
6∆y
+ ciy−10fi(x, y − ∆y) + 2fi(x, y − 2∆y)
6∆y, ciy > 0 , (17)
ciy∂yfi = ciy−fi(x, y − 2∆y) + 2fi(x, y − ∆y) − 9fi(x, y)
6∆y
+ ciy10fi(x, y + ∆y) − 2fi(x, y + 2∆y)
6∆y, ciy < 0 . (18)
The same procedures were carried out for the evolution of temperature equation.
From this combination, the accuracy of the FDTLBM is second order in time and
third order in space. The time step used in the computation is varied between 0.1
and 0.001, depending on the Rayleigh number and mesh size.
4. Natural Convection in a Square Cavity
Numerical simulation for the natural convection flow in a square cavity with a
hot wall on the left side and cool wall on the right side up to Rayleigh number,
Ra = 106 was carried out to test the effectiveness of the FDTLBM. Figure 2 shows
a schematic diagram of the setup in the simulation.
The conventional no-slip boundary conditions1 are imposed on all the walls of
the cavity. The thermal conditions applied on the left and right walls are T (x =
0, y) = TH and T (x = L, y) = TC . The top and bottom walls being adiabatic,
∂T/∂y = 0.
The temperature difference between the left and right walls introduces a tem-
perature gradient in a fluid, and the consequent density difference induces a fluid
motion, that is, convection.
In the simulation, the Boussinesq approximation is applied to the buoyancy
force term.
ρG = ρβg0(T − Tm)j , (19)
-
September 3, 2008 10:0 WSPC/140-IJMPB 04861
3870 C. S. Nor Azwadi & T. Tanahashi
0,0 =∂
∂=
y
Tu
x
y
HTT =
= 0u
CTT =
= 0u
0,0 =∂
∂=
y
Tu
0g
L
L
Fig. 2. Schematic geometry for natural convection in a square cavity.
where β is the thermal expansion coefficient, g0 is the acceleration due to gravity,
Tm is the average temperature, and j is the vertical direction opposite to that of
gravity. So the external force in Eq. (1) will be
Ff = 3G(c − u)f eq . (20)
The dynamical similarity depends on two dimensionless parameters: the Prandtl
number, Pr and the Rayleigh number, Ra
Pr =ν
χ, Ra =
g0β∆TL3
νχ. (21)
In all simulations, Pr is set to be 0.71 and through the grid dependence study,
the grid sizes of 101 × 101, 151 × 151, 201 × 201, and 251 × 251 are suitable forRayleigh number 103, 104, 105, and 106, respectively. The convergence criterion for
all the cases tested is
Max|(
(u2 + v2)n+1)
1
2 −(
(u2 + v2)n)
1
2 | ≤ 10−7 , (22)
= Max|T n+1 − T n| ≤ 10−7 , (23)
where the calculation is carried out over the entire system.
5. Numerical Results
Figures 3–6 show the time development of isotherms and their corresponding
streamlines for all the Rayleigh numbers simulations.
At the beginning of the simulation for Ra = 103, a vortex appears at the center
left of the cavity. As time evolves, the vortex is shifted to the center of the cavity.
The isotherms are almost vertically parallel to the wall, indicating that conduction
is the dominant heat transfer mechanism. For Ra = 104, a vertically oval-shaped
-
September 3, 2008 10:0 WSPC/140-IJMPB 04861
Finite Difference Thermal Lattice Boltzmann Method 3871
Fig. 3. Time development of isotherms and streamlines for Ra = 103.
Fig. 4. Time development of isotherms and streamlines for Ra = 104.
-
September 3, 2008 10:0 WSPC/140-IJMPB 04861
3872 C. S. Nor Azwadi & T. Tanahashi
Fig. 5. Time development of isotherms and streamlines for Ra = 105.
Fig. 6. Time development of isotherms and streamlines for Ra = 106.
-
September 3, 2008 10:0 WSPC/140-IJMPB 04861
Finite Difference Thermal Lattice Boltzmann Method 3873
vortex appears at the center left of the cavity. After that, the vortex is shifted
to the center of the cavity and its shape changes to horizontal oval due to the
convection effect. Isotherms start to be horizontally parallel to the wall at the cavity
center. This indicates that the heat transfer mechanisms are mixed conduction and
convection.
For the simulation at Ra = 105, two vortices appear, where one at the top
left and the other one at the bottom right of the cavity when the system achieved
equilibrium condition. All isotherms are almost horizontally parallel to the wall,
indicating that the convection is the main heat transfer mechanism. The vortices
continue to break up when the Rayleigh number is increased up to 106.
Figure 7 shows the non-dimensional temperature profile given at the mid-height
of the cavity for the laminar flow simulations. The profiles show the rapid change in
the heat transfer mechanisms from conduction to convection. From a 45◦ slope at
low Rayleigh number, the temperature profiles become horizontal lines in the cavity
center and all temperature gradients are located in the interior of the boundary
layer, which has developed near the vertical walls. Near the center of the cavity, the
curves change slope and there is a vortex corresponding to each change. It can be
clearly seen that the steep variation of the temperature near the walls is resolved
quite well.
Figures 8 and 9 present similar profiles for the horizontal and vertical velocity
components, respectively. Both figures show a gradually increasing velocity near
the center, and the development of narrow boundary layers along the walls. The
peak values of the horizontal and vertical velocities increase due to the intensified
convective activities with increase in Rayleigh number. The steep rise in the vertical
velocity gradient at the point on the hot and cold walls also confirms the increased
310=Ra
410=Ra
610=Ra
510=Ra
Fig. 7. Non-dimensional temperature profile at the mid-height of the cavity.
-
September 3, 2008 10:0 WSPC/140-IJMPB 04861
3874 C. S. Nor Azwadi & T. Tanahashi
310=Ra
410=Ra
610=Ra
510=Ra
Fig. 8. Non-dimensional horizontal velocity at the mid-width of the cavity.
310=Ra
410=Ra
610=Ra
510=Ra
310=Ra
410=Ra
610=Ra
510=Ra
Fig. 9. Non-dimensional vertical velocity at the mid-width of the cavity.
convective activity at higher Rayleigh number values observed in Fig. 9. The changes
in the velocity direction correspond to slope changes of the temperature profile and
lead to vortex development.
In order to validate the present numerical algorithm, the predicted results
are compared with the results obtained by the Navier–Stokes equation approach.
Among the characteristic numerical values of the flow, the comparisons concern the
average Nusselt number at the mid-plane wall, Nuave the maximum value for hor-
izontal and vertical velocity components, umax and vmax with the positions where
they occur.
-
September 3, 2008 10:0 WSPC/140-IJMPB 04861
Finite Difference Thermal Lattice Boltzmann Method 3875
Table 1. Comparison between the present results (FDTLBM) and aNavier–Stokes solver.18
Ra
103 104 105 106
umax FDTLBM 3.638 16.127 34.700 65.827
N–S Solver18 3.634 16.182 34.810 65.330
y FDTLBM 0.810 0.820 0.855 0.852
N–S Solver18 0.813 0.823 0.855 0.848
vmax FDTLBM 3.691 19.584 68.319 221.071
N–S Solver18 3.679 19.509 68.220 216.750
x FDTLBM 0.180 0.120 0.065 0.040
N–S Solver18 0.179 0.120 0.066 0.038
Nuave FDTLBM 1.117 2.233 4.483 8.723
N–S Solver18 1.116 2.234 4.510 8.798
As shown in Table 1, for Ra = 103 and Ra = 104, the present results are in
close agreement with the Navier–Stokes solution obtained by Davis.18 However,
small discrepancies can be seen for higher values of Rayleigh numbers, Ra = 105
and Ra = 106. For all values of Rayleigh number considered in the present analysis,
the average Nusselt number for the system have been predicted with less than 3%
error and can be accepted for real engineering applications.
6. Conclusion
The natural convection in a differentially heated square cavity has been studied
using the double-distribution approach thermal lattice Boltzmann method with
small microvelocity components applied in the internal energy distribution func-
tion. The evolution of lattice Boltzmann equations have been discretized using the
third-order accuracy finite difference upwind, UTOPIA scheme. From Figs. 7–9,
the boundary layers for the velocities and temperature can be observed clearly. As
expected, the thermal boundary layer is thicker than the velocity boundary layer
for every Rayleigh number simulations. The flow patterns including the boundary
layers and vortices can be seen clearly. The results obtained demonstrate that this
new approach in the double-distribution function thermal lattice Boltzmann model
is a very efficient procedure to study flow and heat transfer in a differentially heated
square enclosure.
Acknowledgments
The authors wish to acknowledge Universiti Teknologi Malaysia, Keio University
and the Malaysian government for supporting these research activities.
References
1. J. Hardy, Y. Pomeau and D. Pazzis, J. Math. Phys. 14, 1746 (1973).
-
September 3, 2008 10:0 WSPC/140-IJMPB 04861
3876 C. S. Nor Azwadi & T. Tanahashi
2. G. McNamara and B. Alder, Phys. A. 194, 218 (1993).3. X. Shan, Phys. Rev. E. 55, 2780 (1997).4. X. He, S. Shan and G. D. Doolen, J. Comp. Phys. 146, 282 (1998).5. H. Chen and C. Teixeira, Comp. Phys. Comm. 129, 21 (2000).6. Y. Peng, C. Shu and Y. T. Chew, Phys. Rev. E 68, 020671 (2003).7. L. S. Luo and X. He, Phys. Rev. E 55, R6333 (1997).8. X. He and L. S. Luo, Phys. Rev. E 56, 6811282 (1997).9. T. Abe, J. Comp. Phys. 131, 241 (1997).
10. M. B. Reider and J. D. Sterling, Comp. Fluids 24, 459 (1995).11. N. Cao, S. Chen, S. Jin and D. Martinez, Phys. Rev. E 55, R21 (1997).12. Z. R. Mei and W. Shyy, J. Comp. Phys. 143, 426 (1998).13. J. Tolke, M. Krafczvk, M. Schulz, E. Rank and R. Berrios, Int. J. Mod. Phys. C 9,
1143 (1998).14. G. Hazi, Int. J. Mod. C 13, 67 (1998).15. T. Seta and R. Takahashi, J. Stat. Phys. 107, 557 (2002).16. C. S. Nor Azwadi and T. Takahashi, Int. J. Mod. Phys. B 20, 2437 (2006).17. Z. Guo, Y. Shi and T. S. Zhao, Phys. Rev. E 70, 066310 (2004).18. D. V. Davis, Int. J. Numer. Meth. Fluids 3, 249 (1983).